SGR1806-20 evidence for a superstrong Magnetic Field from Quasi Periodic Oscillations

a r X i v :a s t r o -p h /0702598v 1 22 F eb 2007SGR1806-20:evidence for a superstrong Magnetic Field from QuasiPeriodic OscillationsMario VietriScuola Normale Superiore,56100Pisa,Italym.vietri@sns.it and Luigi Stella and Gian Luca Israel INAF -Osservatorio Astronomico di Roma,Via Frascati 33,00040Monteporzio Catone,Italy.stella@mporzio.astro.it gianluca@mporzio.astro.it ABSTRACT Fast Quasi-Periodic Oscillations (QPOs,frequencies of ∼20−1840Hz)have been recently discovered in the ringing tail of giant flares from Soft Gamma Repeaters (SGRs),when the luminosity was of order 1041−1041.5erg/s.These oscillations per-sisted for many tens of seconds,remained coherent for up to hundreds of cycles and were observed over a wide range of rotational phases of the neutron stars believed to host SGRs.Therefore these QPOs must have originated from a compact,virtuallynon-expanding region inside the star’s magnetosphere,emitting with a very moder-ate degree of beaming (if at all).The fastest QPOs imply a luminosity variation of ∆L /∆t ≃6×1043erg s −2,the largest luminosity variation ever observed from a com-pact source.It exceeds by over an order of magnitude the usual Cavallo-Fabian-Rees (CFR)luminosity variability limit for a matter-to-radiation conversion efficiency of 100%.We show that such an extreme variability can be reconciled with the CFR limit if the emitting region is immersed in a magnetic field 1015G at the star surface,providing independent evidence for the superstrong magnetic fields of magnetars.Subject headings:stars:magnetic fields —stars:neutron —stars:individual(SGR 1806-20)1.IntroductionSoft Gamma Repeaters are a small class of galactic sources of X and soft gamma radiation. They have spin periods of∼5÷10s,display a secular spin-down with timescales of∼104÷105yr and do not possess a companion.Unlike radio pulsars,the rotational energy loss of SGRs is a factor of10÷100too small to explain their persistent emission,typically∼1033÷1034erg/s(see e.g. Woods&Thompson(2006)).Like Anomalous X-ray Pulsar(AXPs,Mereghetti&Stella(1995)), with whom they share a number of properties,SGRs are believed to host magnetars,neutron stars the emission of which is powered by the decay of their superstrong(internal)magneticfield(B> 1015G,Duncan&Thompson(1992);Thompson&Duncan(1993)).The name-defining characteristic of SGRs is that they show periods of activity in which re-current short bursts are emitted,with peak luminosities of∼1038÷1041ergs s−1and sub-second durations.The characteristics of the three giantflares observed so far in about30yr of mon-itoring are much more extreme.Their initial,tenths-of-seconds-long spike releases enormous amounts of energy,∼1044ergs in the1979March5event from SGR0526-66(Mazets et al. 1979)and the1998August27event from SGR1900+14(Hurley et al.1999;Feroci et al.1999) and as much as5×1046erg in the2004December27event from SGR1806-20(Hurley et al.2005; Mereghetti et al.2005;Palmer et al.2005;Terasawa et al.2005).After the initial spike,giantflares display a very bright ringing tail lasting hundreds of seconds and releasing a total energy of about ∼1044ergs.The emitted spectrum is roughly thermal,with a blackbody equivalent temperature of ∼5keV in the case of SGR1806-20Hurley et al.(2005).The highly super-Eddington luminosities of the recurrent bursts of SGRs and especially of their giantflares make models involving accretion energy not viable.According to the magne-tar model,the emission of SGRs(and AXPs)draws from their extremely high magneticfields (Thompson&Duncan1995,1996,2001).Within this model the neutron star interior is character-ized by a wound-up,mainly toroidal magneticfield configuration with B t>1015G.A less intense (mainly poloidal)field emerges out of the star magnetosphere,causing spin-down via rotating dipole losses at the observed rate(Thompson&Duncan1993,2001)(dipole B-field strength of B d∼7.8×1014G).Energy propagates to the neutron star magnetosphere through Alfvén waves driven by local“crust-quakes"and giving rise to recurrent bursts with a large range of amplitudes. Large-scale rearrangements of the internal magneticfield or catastrophic instabilities in the mag-netosphere are invoked to explain the sudden release of very large amounts of energy that occurs in giantflares(Thompson&Duncan2001;Lyutikov2003).Afireball of plasma expanding at relativistic speeds breaks out of the star’s magnetosphere,causing the initial sub-second spike of giantflares.The ringing tail that follows results most likely from the part of thefireball that re-mains trapped in the star’s magnetosphere.The energy release in the ringing tail yields a limit for the externalfield of magnetars(>∼1014G)in agreement with the values inferred from spin-downdipole losses(Thompson&Duncan1995,2001),while an analysis of the initial time scales gives evidence in favor of the crustal cracking mechanism(Schwartz et al.2005).To confirm this model it is thus essential to measure the surface magneticfield,which com-plements the measurement of the dipole component(Woods et al.(2002)),to include higher order multipoles.In this paper we present an interpretation of the QPOs observed in SGR1806-20which provides a lower limit(B≫B q≈4.4×1013G)to the surfacefield.The argument is based upon a seldom-used constraint((Cavallo&Rees1978;Fabian&Rees1979;Fabian1979))that puts a very strong upper limit on the time-scale on which significant luminosity variations can take place. In the next section we discuss the relevant observations;in Section3we will re-derive the limit and show that it is largely violated by QPOs in SGR1806-20;after vainly trying to circumvent it, we will show that it can be reconciled with observations only if B≫B q.2.Quasi Periodic Oscillations in Giant Flares of SGRsRecent studies led to the discovery that the X-rayflux of the ringing tail of SGRs’giant flares is characterized by fast Quasi Periodic Oscillations,QPOs(Israel et al.2005).Different QPO modes were detected,some of which were excited simultaneously.The ringing tail of the December2004event from SGR1806-20displayed clear QPO signals at about18,30,93,150, 625and1840Hz(Strohmayer&Watts(2006a)).Similarly,QPOs around frequencies of28,54, 84and155Hz were detected during the ringing tail of the1998giantflare of SGR1900+14 (Strohmayer&Watts2005),while hints for a signal at∼43Hz were found in the March1979 event from SGR0526-66(Barat et al.1983).These QPOs show large variations of the amplitude with time and,especially,of the phase of the spin modulation in the giantflare’s tail.The similarity in some of the QPO modes and frequencies across different SGRs suggests that the production mechanism is the same.A likely interpretation involves the excitation of neutron star oscillation modes,whose expected eigenfrequencies match some of the observed QPOs peaks ((Duncan1998;Israel et al.2005;Piro2005)).Infact,if giantflares result from large scale frac-turing of the crust induced by instabilities of the internal magneticfields,then the excitation of crustal and(possibly)global neutron star modes is to be expected((Levin2006)).Regardless of the exact mechanism driving these oscillations we are concerned here with the extremely large and fast luminosity variations of the QPOs.We concentrate on the signals with the largest luminosity time derivative,i.e.the625and 1840Hz QPOs from SGR1806-20(Strohmayer&Watts(2006a)).The power spectrum peaks through which these QPOs are revealed,are a few Hz wide,testifying that their signal remained coherent for hundreds of cycles.The signal shape must be close to sinusoidal,as evidenced bythe absence of detectable harmonic signals.The625and1840Hz QPOs were detected only in a∼50s long interval of the ringing tail,about200s after the initial spike,and were especially prominent over a∼140deg interval in rotational phase.The QPO amplitude reached a maximum in this phase interval over two consecutive rotation cycles:for both signals the rms amplitude was a rms∼18%.Approximating the QPOs with sinusoids,we estimate their highest luminosity derivative as∆L/∆t=23/2πLa rmsνQPO,withνQPO the QPO frequency.Here L∼1041ergs s−1is the luminosity in the relevant section of the ringing tail(for the likely source distance of15kpc). This gives∆L/∆t∼1×1044and3×1044ergs s−2for the625and1840Hz QPOs,respectively. The effects of beaming might decrease these values somewhat,but not by a very large factor.In fact,these QPO signals were observed over a large interval of rotational phases(about∼140deg), translating into approximately the same azimuthal range of emission angles.It is natural to assume a comparably large angular spread in latitude(unless the neutron star rotation axis is very close to our line of sight,which is unlikely given the large amplitude of the rotation modulation and the size of the emission region,see below).This gives a solid angle of order∼πster.Adopting this beaming factor the luminosity derivatives above reduce to∼2×1043and6×1043ergs s−2;this are the values that we adopt in the following discussion.We stress here,because this is essential to our argument(to be presented shortly),that,to-gether with the QPOs,in the ringing tail a strong modulation at the star’s spin period is clearly present,similar in relative amplitude and shape to the modulation observed when the source is in its quiescent state:this(together with the lack of significant amounts of beaming)indicates that the emission in the ringing tail originates from a region that remains stably anchored to the star’s magnetosphere,and thus that relativistic bulk motions are not present at this stage of theflare.The blackbody temperature and luminosity in the ringing tail translates into a lower limit on the size of the emitting region of about∼30km,i.e.substantially larger than the neutron star.On the other hand the black body-like spectral shape testifies that the emitting region is optically thick (or at least effectively thick),implying a scattering optical depth≫1.We remark that the size estimate,≈30km,will play an important role in the following.3.The Cavallo-Fabian-Rees Variability LimitThere is a well–known limit on the rate of change of the luminosity of any given source, which we briefly summarize here(Cavallo&Rees(1978);Fabian&Rees(1979);Fabian(1979), see also Lawrence(1980);Hoshi(1989)).Suppose a source undergoes a large luminosity variation on a time-scale∆t,and there is a luminosity variation,over this time-scale,∆L.The total energyreleased within∆t is related to the total mass within the source dimension R by∆L∆t=η4πc(1+τT)(2) whereτT=σT nR is the Thomson optical depth andσT the Thomson cross-section.Eliminating R from thefirst two equations,wefind∆t>3m p c4∆LτT.(3)Regarded as a function ofτT,the above has a minimum forτT=1,which yields the limit∆t>3m p c4∆L1We note that also the slower∼90−150Hz QPOs signals in the giantflares’tail of SGR1806-20and SGR1900+14, exceed the CFR limit,though by a smaller factor.the more sedateηm p c2.This however appears like an unlikely way out,because,even admitting that the electrons radiate very promptly their internal energy,the time-scale on which protons manage to transfer to the electrons their internal energy is much longer than the above limit.To see this,we idealize the situation as one where electrons are cold(i.e.Newtonian)as a result of their short cooling timescales,while the protons are still relativistic.The energy transfer rate is given by the usual formula−dE pm e v2 ln(2m e v2W c2,(6)where n e is the electron number density,ωp=ln(2m e v2W m p c4Lm p m e c3.(7)Inserting the numerical values for the luminosity of SGR1806-20during the ringing tail and for the typical magnetospheric density for a pulsar with a normal magneticfield wefindγ 10−813×1041ergs−1n e3R3nm e c2(9)where we used the fact thatγ≈2for the electrons.The new limit for the variability is∆t>3m e c4∆Lη∆L4σT sin2θsin2θ′d cosθ′dσO→E=3m e c2B 2cos2θd cosθ′dσE→O=3m e c2B 2cos2θ′d cosθ′dσE→E=3m e c2B 2d cosθ′,(11)whereǫis the photon energy,andθandθ′are the angles between the photon momentum before and after the diffusion,respectively,with the direction of the magneticfield.This equation shows immediately that,for photons emitted in the extraordinary mode,the cross section is reduced,with respect to the Thomson value,by a factor≈(ǫB q/(m e c2B))2.Thus, to bring the observed value,△L/△t=6×1043erg s−1,in agreement with eq.4,we just need to have(ǫB q/(m e c2B))2 η/30;here we take forǫthe valueǫ≈14keV,which is the peak of the Planck distribution for the observed temperature T=5keV.So our conclusion is that the QPOs’luminosity variation agrees with the CFR limit providedB 1.5B q 0.1R ns 30.14.DiscussionOurfinal result is the lower limit in eq.13.This value is close to that of the dipolefield,as inferred from pulsar spin-down,B=7.8×1014G,but our estimate includes higher order multipoles, at least at30km.It is thus completely independent of the estimates from pulsar spin-down.We note that the limit on B derived by requiring that the magneticfield close to the star can prevent the escape of the"trappedfireball"includes also the contribution of higher multipole components (Thompson&Duncan(2001)),but gives a substantially lower magneticfield(B>1014G)than the limit derived above.There is an easy way to test our interpretation of the failure of the CFR limit.The O-and E-mode photospheres of the trappedfireball are located at different heights in the star’s magneto-sphere,as a result of the different electron scattering cross sections,which therefore sample regions of different B-field strengths and orientations.The polarization fraction and angle of the emerging X-rayflux should thus be modulated with the phase of the QPO signal as a result of the varying (relative)intensity of the O-and E-mode photon component.This in principle gives a clear test for the correctness of our interpretation,which might perhaps become verifiable in the future.Photons in the O-mode suffer a strong Comptonization(Thompson&Duncan(2001)).If the atmosphere were due to pure scattering,the ensuing photon distribution would differ from a blackbody at low photon frequency,Eν∝ν3instead of∝ν2,a result due to photon number con-servation.Atfirst sight,one might think that there are many processes which may lead to photon absorption and emission in a strong magneticfield,like photon splitting or pair creation viaγ+B (resulting then in pair annihilation,and photon energy downgrading via Compton recoil),which make photon number conservation unlikely.At the same time,there are important radiation trans-fer effects taking place,which obviously tend to favorflatter spectra at low energies;also,all rates for photon emission and absorption depart from their values in the absence of magneticfield.De-tailed computations((Lyubarsky2002))show that the ensuing spectrum isflatter,not steeper,than a blackbody;and we remark here that,despite many calibration uncertainties,fits to the spectrum seem to favor spectraflatter than a blackbody,with a preference for thermal bremsstrahlung of temperature T=30keV.We have presented our argument by stressing its independence from the estimates of the dipole field from pulsar spin-down,and from Thompson and Duncan’s(Thompson&Duncan(1993, 1995,1996))model,but it should be obvious that the limit in eq.13is perfectly consistent with the previous measurements,and the model itself.This work was partially supported through ASI and MIUR grants.REFERENCESBarat,C.et al.1983,A&A,126,400Bassani,L.,Dean,A.J.&Sembay,S.1983,A&A,125,52Cavallo,G.&Rees,M.1978,MNRAS,125,52Duncan,R.C.1998ApJ,498,L45Duncan,R.,C.,&Thompson,C.1992,ApJ,392,L9Fabian,A.C.1979,Proc.R.Soc.London,Ser.A.,336,449Fabian,A.C.&Rees,M.1979,Adv.Sp.Res.,3,381Feroci,M.,et al.1999,ApJ,515,L9Hurley,K.,et al.1979,Nature,397,41Hurley,K.,et al.2005,Nature,434,1098Mazets,E.,P.,Golentskii,S.,V.,Ilinskii,V.,N.,Aptekar,R.,L.,Guryan&Iu.,A.1979,Nature, 282,587Hoshi,R.1989,PASJ,41,217Israel,G.L.et al.2005,ApJ,628,L53Lai,D&Ho,W.C.G.,2003,ApJ,588,962.Lawrence,A.1980,MNRAS,192,83Levin,Y.2006,MNRAS,368,L35Lyubarsky,Y.E.,2002,MNRAS,332,199.Lyutikov,M.,2003,MNRAS,346,540Mereghetti,S.,&Stella,L.1995,ApJ,442,L17Mereghetti,S.,Götz,D.,von Kienlin,A.,Rau,A.,Lichti,G.,Weidenspointner,G.&Jean,P.2005, ApJ,624,L105Meszaros,P.,1992,High energy radiation from magnetized neutron stars,Univ.Chicago Press, Chicago.–11–Palmer,D.M.,et al.2005,Nature,434,1107Piran,T.,2005,Rev.Mod.Phys.,76,1143Piro,A.L.2005ApJ,634,L153Pietrini,P.Krolik,J.H.,2005,ApJ,447,526.Schwartz,S.J.,et al.,2005,ApJ,627,L129.Strohmayer,T.E.,&Watts,A.L.2005,ApJ,632,L111.Strohmayer,T.E.&Watts,A.L.,2006,ApJ,653,593.Terasawa,T.,et al.2005,Nature,434,1110Thompson,C.,&Duncan,R.,C.1993,ApJ,408,194Thompson,C.,&Duncan,R.C.1995,MNRAS,275,255Thompson,C.,&Duncan,R.C.1996,ApJ,473,322Thompson,C.,&Duncan,R.C.2001,ApJ,561,980Woods,P.M.et al.ApJ,576,381Woods,P.M.,&Thompson,C.2006,in"Compact Stellar X-ray Sources"Ed.Walter Lewin, W.H.G.,van der Klis,M.(Cambridge University Press),p.547.。